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A384978 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384975.

%I A384978 #10 Jun 14 2025 10:36:42
%S A384978 1,1,0,1,1,0,1,2,4,0,1,3,9,16,0,1,4,15,40,77,0,1,5,22,73,202,303,0,1,
%T A384978 6,30,116,387,888,718,0,1,7,39,170,645,1851,2914,-4934,0,1,8,49,236,
%U A384978 990,3304,7267,-3544,-108553,0,1,9,60,315,1437,5376,14616,8463,-205605,-1275290,0
%N A384978 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384975.
%F A384978 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-5*n+5*j+k-1,j-1) * b(n-j,3*j)/j. Then A(n,k) = b(n,-k).
%e A384978 Square array begins:
%e A384978   1,   1,    1,    1,     1,     1,     1, ...
%e A384978   0,   1,    2,    3,     4,     5,     6, ...
%e A384978   0,   4,    9,   15,    22,    30,    39, ...
%e A384978   0,  16,   40,   73,   116,   170,   236, ...
%e A384978   0,  77,  202,  387,   645,   990,  1437, ...
%e A384978   0, 303,  888, 1851,  3304,  5376,  8214, ...
%e A384978   0, 718, 2914, 7267, 14616, 25980, 42579, ...
%o A384978 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-5*n+5*j+k-1, j-1)*b(n-j, 3*j)/j));
%o A384978 a(n, k) = b(n, -k);
%Y A384978 Columns k=0..1 give A000007, A384975.
%K A384978 sign,tabl
%O A384978 0,8
%A A384978 _Seiichi Manyama_, Jun 14 2025